On the Block Coloring of Steiner Triple Systems (original) (raw)

Equitable Specialized Block-Colourings for Steiner Triple Systems

Graphs and Combinatorics, 2008

We continue the study of specialized block-colourings of Steiner triple systems initiated in in which the triples through any element are coloured according to a given partition π of the replication number. Such colourings are equitable if π is an equitable partition (i.e., the difference between any two parts of π is at most one). Our main results deal with colourings according to equitable partitions into two, and three parts, respectively.

Uniquely 3-colourable Steiner triple systems

Journal of Combinatorial Theory, Series A, 2003

A Steiner triple system ðSTSðvÞÞ is said to be 3-balanced if every 3-colouring of it is equitable; that is, if the cardinalities of the colour classes differ by at most one. A 3-colouring, f; of an STSðvÞ is unique if there is no other way of 3-colouring the STSðvÞ except possibly by permuting the colours of f: We show that for every admissible vX25; there exists a 3-balanced STSðvÞ with a unique 3-colouring and an STSðvÞ which has a unique, non-equitable 3colouring.

Extended Bicolorings of Steiner Triple Systems of Order 2h−12^{h}-12h−1

Taiwanese Journal of Mathematics, 2017

A bicoloring of a Steiner triple system STS(n) on n vertices is a coloring of vertices in such a way that every block receives precisely two colors. The maximum (resp. minimum) number of colors in a bicoloring of an STS(n) is denoted by χ (resp. χ). All bicolorable STS(2 h − 1)s have upper chromatic number χ ≤ h; also, if χ = h < 10, then lower and upper chromatic numbers coincide, namely, χ = χ = h. In 2003, M. Gionfriddo conjectured that this equality holds whenever χ = h ≥ 2. In this paper we discuss some extensions of bicolorings of STS(v) to bicoloring of STS(2v + 1) obtained by using the 'doubling plus one construction'. We prove several necessary conditions for bicolorings of STS(2v + 1) provided that no new color is used. In addition, for any natural number h we determine a triple system STS(2 h+1 − 1) which admits no extended bicolorings.

Colouring Steiner systems with specified block colour patterns

Discrete Mathematics, 2001

We consider colourings of Steiner systems S(2; 3; v) and S(2; 4; v) in which blocks have prescribed colour patterns, as a reÿnement of the classical weak colourings. The main question studied is, given an integer k, does there exist a colouring of given type using exactly k colours? For several types of colourings, a complete answer to this question is obtained while for other types, partial results are presented. We also discuss the question of the existence of uncolourable systems.

Appendix of Extending Bicoloring for Steiner Triple Systems

We initiate the study of extended bicolorings of Steiner triple systems (STS) which start with a k-bicoloring of an STS(v) and end up with a kbicoloring of an STS(2v+1) obtained by a doubling construction, using only the original colors used in coloring the subsystem STS(v). By producing many such extended bicolorings, we obtain several infinite classes of orders for which there exist STSs with different lower and upper chromatic number

Strict colorings of Steiner triple and quadruple systems: a survey

Discrete Mathematics, 2003

The paper surveys problems, results and methods concerning the coloring of Steiner Triple and Quadruple Systems viewed as mixed hypergraphs. In this setting, two types of conditions are considered: each block of the Steiner system in question has to contain (i) a monochromatic pair of vertices, or, more restrictively, (ii) a triple of vertices that meets precisely two color classes.

Colouring Cubic Graphs by Small Steiner Triple Systems

Graphs and Combinatorics, 2007

Given a Steiner triple system S, we say that a cubic graph G is S-colourable if its edges can be coloured by points of S in such way that the colours of any three edges meeting at a vertex form a triple of S. We prove that there is Steiner triple system U of order 21 which is universal in the sense that every simple cubic graph is U-colourable. This improves the result of Grannell et al. [J. Graph Theory 46 (2004), 15-24] who found a similar system of order 381. On the other hand, it is known that any universal Steiner triple system must have order at least 13, and it has been conjectured that this bound is sharp (Holroyd andŠkoviera [J. Combin. Theory Ser. B 91 (2004), 57-66]).

Extending bicolorings for Steiner triple systems

Applicable Analysis and Discrete Mathematics, 2013

A BSTS(2v + 1) containing a colorable subsystem BSTS(v) with hcoloring C ′ has an extended h-coloring of C ′ if it is also h-colorable with a coloring C in which the subsystem BSTS(v) is colored with C ′ . In this paper we give both necessary conditions and sufficient conditions for the existence of an extended coloring. The existence of these colorings is studied either for systems of type BSTSs(2v + 1) containing subsystems BSTSs(v) with 2v + 1 < 103 or systems of type BSTSs(2 h + 1).

Caps and colouring Steiner triple systems

Designs, Codes and Cryptography, 1998

showed that the largest cap in PG(5, 3) has cardinality 56. Using this cap it is easy to construct a cap of cardinality 45 in AG(5, 3). Here we show that the size of a cap in AG(5, 3) is bounded above by 48. We also give an example of three disjoint 45-caps in AG(5, 3). Using these two results we are able to prove that the Steiner triple system AG(5, 3) is 6-chromatic, and so we exhibit the first specific example of a 6-chromatic Steiner triple system.

Colouring of cubic graphs by Steiner triple systems

Journal of Combinatorial Theory, Series B, 2004

Let S be a Steiner triple system and G a cubic graph. We say that G is S-colourable if its edges can be coloured so that at each vertex the incident colours form a triple of S. We show that if S is a projective system P G(n, 2), n ≥ 2, then G is S-colourable if and only if it is bridgeless, and that every bridgeless cubic graph has an S-colouring for every Steiner triple system of order greater than 3. We establish a condition on a cubic graph with a bridge which ensures that it fails to have an S-colouring if S is an affine system, and we conjecture that this is the only obstruction to colouring any cubic graph with any non-projective system of order greater than 3.